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Question: The demand function is \(x=\dfrac{24-2p}{3}\) , where x is the number of units demanded and p is the...

The demand function is x=24−2p3x=\dfrac{24-2p}{3} , where x is the number of units demanded and p is the price per unit. Find:
(i) The revenue function R in terms of p.
(ii) The price and the number of units demanded for which the revenue is maximum.

Explanation

Solution

  1. Revenue = Price per unit × Number of units demanded.
  2. The maximum/minimum value of a function f(x) is attained when f’(x) = 0.
  3. At the points of maxima, f’’(x) is < 0 and at the points of minima, f’’(x) > 0.
  4. Differentiation: ddxxn=nxn−1\dfrac{d}{dx}{{x}^{n}}=n{{x}^{n-1}} .

Complete step by step solution:
Given that Number of units = x and price per unit = p.
(i) Using the formula, Revenue = Price per unit × Number of units demanded, we will get:
Revenue=x×p=(24−2p3)×p=8p−2p23\text{Revenue}=x\times p=\left( \dfrac{24-2p}{3} \right)\times p=8p-\dfrac{2{{p}^{2}}}{3} , which is the required revenue function in terms of p.
(ii) Since, the revenue function is in terms of p, let us differentiate it with respect to p and equate it to 0 to find the points of maxima/minima.
ddp(8p−2p23)=0\dfrac{d}{dp}\left( 8p-\dfrac{2{{p}^{2}}}{3} \right)=0
On differentiating, we get

⇒ 8−2×2p3=08-\dfrac{2\times 2p}{3}=0
After taking LCM and simplify, we get
⇒ 4p3=8\dfrac{4p}{3}=8
⇒ p = 6.
Also, d2dp2(8p−2p23)=ddp[ddp(8p−2p23)]=ddp(8−4p3)=−43\dfrac{{{d}^{2}}}{d{{p}^{2}}}\left( 8p-\dfrac{2{{p}^{2}}}{3} \right)=\dfrac{d}{dp}\left[ \dfrac{d}{dp}\left( 8p-\dfrac{2{{p}^{2}}}{3} \right) \right]=\dfrac{d}{dp}\left( \dfrac{8-4p}{3} \right)=\dfrac{-4}{3} .
Since, the second derivative is < 0, the value of the revenue function at p = 6 is maximum.
And, using the given relation, x=24−2p3=24−2×63=123=4x=\dfrac{24-2p}{3}=\dfrac{24-2\times 6}{3}=\dfrac{12}{3}=4 .
The maximum value of the revenue is obtained for x = 4 units and price p = 6 per unit.

Note:

  1. Demand function shows the relationship between quantity demanded for a particular commodity and the factors that are influencing it (like price of the commodity, price of related goods, income of consumer etc).
  2. The maximum/minimum value of a quadratic function can also be found by the method of completion of squares or by using the fact that the maximum/minimum value of a quadratic function is obtained at the mean of its roots.